2000 years unsolved: Why is doubling cubes and squaring circles impossible?

From Mathologer.

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Today’s video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles?

Towards the end of a pure maths degree students often have to survive a "boss" course on Galois theory and somewhere in this course they are presented with proofs that it is actually not possible to accomplish any of those four troublesome tasks. These proofs are easy consequences of the very general tools that are developed in Galois theory. However, taken in isolation, it is actually possible to present proofs that don’t require much apart from a certain familiarity with simple proofs by contradiction of the type used to show that numbers like root 2 are irrational.

I’ve been meaning to publish a nice exposition of these "simple" proofs ever since my own Galois theory days (a long, long time ago.
Finally, today is the day 🙂

For some more background reading I recommend:
1. chapter 3 of the book "What is mathematics?" by Courant and Robbins (in general this is a great book and a must read for anybody interested in beautiful maths).
2. The textbook "Field theory and its classical problems" by Hadlock (everything I talk about and much more, but you need a fairly strong background in maths for this one).

Here is a great two-page summary by the mathematician Drew Armstrong of what is going on in this video
http://www.math.miami.edu/~armstrong/461sp11/ImpossibleConstructions.pdf

Here is a derivation of the cubic polynomial for the regular heptagon construction by (I think) the mathematician Reinhard Schultz http://math.ucr.edu/~res/math153/s10/history09a.pdf
(there is a little typo towards the bottom of the page. It should be 8 cos^3 theta + 4 cos^2 theta – (!) 4 cos theta -1 = 0. Replace cos theta by x and you get the cubic equation I mention in the video. )

Here is an interesting paper that explores why Wantzel’s results did not get recognised during his lifetime https://www.sciencedirect.com/science/article/pii/S031508600900010X

Thank you to Marty and Karl for your help with creating this video. And thank you to Cleon Teunissen for pointing out that the picture of Pierre Wantzel that I use in this video is actually not showing Pierre Wantzel but rather Gustave Gaspard de Coriolis who was also a mathematician and lived around the same time as Pierre Wantzel. It appears that whenever there does not exist an actual picture of some person Google and other internet gods simply declare some more or less random picture to be the real thing. See also this page by the SciFi writer Greg Egan whose made sure that no actual picture of himself is to be found on the internet: https://www.gregegan.net/images/GregEgan.htm

Enjoy 🙂

Burkard